Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m=\Delta m-nm,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.$$ in a bounded domain $\Omega\subset \mathbb{R}^3$ with smooth boundary, where $\kappa\in \mathbb{R}$ is given constant, $S$ is a matrix-valued sensitivity satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with some $C_S>0$ and $\alpha\geq 0$. As the case $\kappa = 0$ (with $\alpha\geq\frac{1}{3}$ or the initial data satisfy a certain smallness condition) has been considered in [14], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with $\kappa \neq 0$ admits at least one global weak solution if $\alpha>0$. To the best of our knowledge, this is the first analytical work for the {\bf full three-dimensional four-component} chemotaxis-Navier-Stokes system.


1.
Introduction. Many phenomena, which appear in natural science, especially, biology and physics, support animals' lives (see [46,35,8,24]). Chemotaxis has been extensively studied in the context of modeling mold and bacterial colonies (see Hillen and Painter [10] and Bellomo et al. [1]). In order to describe this biological phenomenon in mathematics, in 1970, Keller and Segel ([15]) proposed the following system n t = ∆n − χ∇ · (n∇c), c t = ∆c − c + n, (1.1) which is called Keller-Segel system. Here χ > 0 is called chemotactic sensitivity, n and c denote the density of the cell population and the concentration of the attracting chemical substance, respectively. Since then, there has been an enormous amount of effort devoted to the possible blow-up and regularity of solutions, as well as the asymptotic behavior and other properties (see e.g. [1]). We refer to [10,11] and [26] for the further reading. Beyond this, a large number of variants of system (1.1) have been investigated, including the system with the logistic terms (see [2,32,39,56], for instance) and the nonlinear diffusion ( [29,45,51,49,50,52]), the signal is consumed by the cells (see e.g. Tao and Winkler [30], [57]) two-species chemotaxis system (see [19,53], for instance) and so on.
In order to discuss of the coral fertilization, Kiselev and Ryzhik ([16] and [17]) investigated the important effect of chemotaxis on the coral fertilization process via the Keller-Segel type system of the form ρ t + u · ∇ρ = ∆ρ − χ∇ · (ρ∇c) − ρ q , 0 = ∆c + ρ, (1.2) where ρ is the density of egg (sperm) gametes, u is the smooth divergence free sea fluid velocity and c denotes the concentration of chemical signal which is released by the eggs. This model (1.2) implicitly assumes that the densities of sperm and egg gametes are identical. Kiselev and Ryzhik ([16] and [17]) proved that if q > 2 and the chemotactic sensitivity χ increases, for the associated 2D Cauchy problem of (1.2), the total mass R 2 ρ can become arbitrarily small, whereas if q = 2, a corresponding weaker but yet relevant effect within finite time intervals is detected (see Kiselev and Ryzhik [17]).
Recently, in order to analyze a further refinement of the model (1.3) which explicitly distinguishes between sperms and eggs, Espejo and Winkler ([6]) proposed the following four-component Keller-Segel(-Navier)-Stokes system with (rotational flux): (1.4) in a domain Ω ⊂ R N (N = 2), where u, P, φ, κ ∈ R and c are defined as before and S is a tensor-valued function or a scalar function which satisfies (1.6) with some C S > 0 and α > 0. Here the scalar functions n = n(x, t) and m = m(x, t) denote the population densities of unfertilized sperms and eggs, respectively. In [6], assuming that S(x, n, c) ≡ 1, Espejo and Winkler showed that the 2D fourcomponent Keller-Segel-Navier-Stokes system (1.4) possesses at least one bounded classical solution, whereas, in three dimensions, Li, Pang and Wang ([18]) showed that the four-component Keller-Segel-Stokes (κ = 0 in the fourth equation of (1.4)) system (1.4) with tensor-valued function (where the tensor-valued function S satisfies (1.6) with α ≥ 1 3 ) possesses at least one bounded classical solution. Recently, by using a (new) weighted estimate, Zheng ([55]) proved that if S satisfies (1.6) with α > 0, the four-component Keller-Segel-Stokes system (1.4) admits at least one bounded classical solution. These indeed extend and improve the recent corresponding results obtained by Li,Pang and Wang ([18]). However, it seems that the method in [55] can not deal with the full three-dimensional four-component chemotaxis-Navier-Stokes system (1.4) (κ = 0 in (1.4)). Moreover, as far as we know, for the full three-dimensional four-component chemotaxis-Navier-Stokes system (1.4) (κ = 0 in (1.4)) it is still not clear whether the solution of the system (1.4) exists or not. Furthermore, in [6], [18] and [55], the authors also showed that the corresponding solutions converge to a spatially homogeneous equilibrium exponentially as t → ∞ as well.
Motivated by the above works, the main objective of the paper is to investigate the four-component Keller-Segel-Navier-Stokes system (1.4) with rotational flux. We sketch here the main ideas and methods used in this article. A key role in our existence analysis is played by the observation that for appropriate positive constants a i and b i (i = 1, 2), the functional 12 possesses a favorable entropy-like property, where n ε , c ε and u ε are components of the solutions to (2.1). This will entail a series of a priori estimates which will derive further ε-independent bounds for spatio-temporal integrals of the approximated solutions and several ε-independent regularity features of their time derivatives (see . On the basis of the compactness properties thereby implied, we shall finally pass to the limit along an adequate sequence of numbers ε = ε j 0 and thereby verify the main results (see Section 6).
Before going into our mathematical analysis, we recall some important progresses on system (1.4) and its variants. In order to describe the behavior of bacteria of the species Bacillus subtilis suspended in sessile water drops, Tuval et al. ( [33]) proposed the following chemotaxis-fluid model where f (c) is the consumption rate of the oxygen by the cells. The model (1.7) occurs in the modelling of the collective behaviour of chemotaxis-driven swimming aerobic bacteria.
As pointed out by Xue and Othmer in [47], the chemotactic sensitivity S should be a tensor function rather than a scalar one, so that, the corresponding chemotaxisfluid system (1.7) loses some energy-like structure, which plays a key role in the analysis of the scalar-valued case. Therefore, there are only a few works concerning chemotaxis-fluid coupled models with tensor-valued sensitivity (see Ishida [12], Wang et al. [9,34], Winkler [42] and Zheng [55] for example).
In comparison to (1.7), if we assume that the signal is produced rather than consumed by cells, then the corresponding chemotaxis-fluid model is the Keller-Segel-fluid system of the form (see [44,38,36,37,54,14]) (1.8) Over the past few years, the mathematical analysis of (1.8) (with tensor-valued sensitivity) began to flourish (see [44,38,36,37,54,14] and references therein ) proved the same result for for the three-dimensional Stokes version (1.8) (i.e., the system with κ = 0). Wang and Liu ([22]) showed that 3D Keller-Segel-Navier-Stokes (κ = 0 in the third equation of (1.8)) system (1.8) admits a global weak solution for tensor-valued sensitivity S(x, n, c) satisfies (1.5) and (1.6) with α > 3 7 . More recently, Ke and Zheng ([14]) extends the result of [22] to the case α > 1 3 , which in light of the known results for the fluid-free system mentioned above is an optimal restriction on α. Some other results on global existence and boundedness properties have also been obtained for the variant of (1.8) obtained on replacing ∆n by nonlinear diffusion operators generalizing the porous medium-type choice ∆n m for several ranges of m > 1 ( [54,25,20,52]).
(1.10) Under these assumptions, our main result can be read as (ii) We should point that the idea of this paper can not deal with the case α = 0, since, it is hard to establish the ε-independent estimates (see the proof of Lemma 4.1). Therefore, when N = 3 and α = 0, we must choose other methods. In addition, we evaluate that the global existence of system (1.4) may depend on the initial data and the C S when α = 0. Here C S is the same as (1.6).
(iii) We have to leave open the question whether the condition (1.11) is optimal or not.

2.
Preliminaries. Due to the strongly nonlinear term κ(u · ∇)u and the presence of tensor-valued S in system (1.4), we need to consider an appropriately regularized problem of (1.4) at first. According to the ideas in [43], the corresponding regularized problem is introduced as follows: and is the standard Yosida approximation. Here (ρ ε ) ε∈(0,1) ∈ C ∞ 0 (Ω) be a family of standard cut-off functions satisfying 0 ≤ ρ ε ≤ 1 in Ω and ρ ε 1 in Ω as ε 0. By an adaptation of well-established fixed point arguments (see e.g. Lemma 2.2 of [43] as well as [42]) and a suitable extensibility criterion, one can readily verify the local solvability of (2.1).
3. Some basic estimates and global existence in the regularized problems.
In this section we want to ensure that the time-local solutions obtained in Lemma 2.1 are in fact global solutions. To this end, in a first step, upon a straightforward integration of the first, two and three equations in (2.1) over Ω, we can establish the following basic estimates by using the maximum principle to the second and third equations. The detailed proof can be found in Lemma 2.2 of [6] (see also [18]). Therefore, we list them here without proof.
Lemma 3.1. There exists λ > 0 independent of ε such that the solution of (2.1) satisfies as well as With all the above estimates at hand, we can now establish the global existence result of our approximate solutions. If you need, refer Lemma 3.9 in [42] for Step 3 and Step 6.

LING LIU, JIASHAN ZHENG AND GUI BAO
As the last summand in (3.26) is nonnegative by the maximum principle, so that, we can thus estimate and λ 1 is the first nonzero eigenvalue of −∆ on Ω under the Neumann boundary condition. And thereby by using the Young inequality. Assume that T max,ε < ∞. In view of (3.13), (3.18) (3.25) and (3.28), we apply Lemma 2.1 to reach a contradiction.

4.
A priori estimates for the regularized problem (2.1) which is independent of ε. Since we want to obtain a weak solution of (1.4) by means of taking ε 0 in (2.1), we will require regularity information which is independent of ε ∈ (0, 1). The main portion of important estimates will be prepared in the following section. Moreover, for T > 0, it holds that one can find a constant C > 0 independent of ε such that Proof. Let p = 4α + 2 3 . We first obtain from ∇ · u ε = 0 in Ω × (0, T max,ε ) and straightforward calculations that for all t > 0. Therefore, in light of (1.6), with the help of the Young inequality, we can estimate the right of (4.3) by following by using the fact that (1 + n ε ) −2α ≤ n −2α ε for all ε ≥ 0, n ε and α ≥ 0. In the following we will estimate the term |p−1| 2 C 2 S Ω n p−2α ε |∇c ε | 2 in the right hand side of (4.4). To this end, we firstly invoke the Gagliardo-Nirenberg inequality again to obtain C 1 > 0 and C 2 > 0 such that L 2 (Ω) + 1) (4.5) by using (3.1) and p = 4α + 2 3 . Next, recalling the Young inequality, , where C 2 is the same as (4.5). Inserting (4.6) into (4.4), we may derive that (4.7) Next, using the Gagliardo-Nirenberg inequality and (3.1), one can get for some positive constants λ 0 , λ 1 and λ 2 independent of ε. Collecting (4.3)-(4.5) and (4.7)-(4.8), we conclude that there exist positive constants C 4 and C 5 such that To track the time evolution of c ε , taking −∆c ε as the test function for the second equation of (2.1) and using (3.1), we have which together with the fact that where λ 2 is the same as (4.8). This together with (4.8) yields to L 2 (Ω) + C 6 (4.12) by using (3.2). Taking an evident linear combination of the inequalities provided by (4.9) and (4.12), we conclude where κ 0 = 4λ 2 C 4 , C 7 = 4C 6 C 4 + C 5 . (4.14) Now, multiplying the fourth equation of (2.1) by u ε , integrating by parts and using ∇ · u ε = 0 Noticing the fact W 1,2 (Ω) → L 6 (Ω) in the 3D case and making use of the Hölder inequality and the Young inequality we can estimate the right hand of (4.15) as where κ 0 is given by (4.14), C 9 , C 10 and C 11 are positive constants which are independent of ε. Here the last inequality we have used the fact that Now, substituting (4.17) into (4.15), one has L 2 (Ω) + C 12 for all t > 0, (4.18) so that, which together with (4.13) implies that Case p > 1. Then sign(p − 1) = 1 > 0. Thus, (4.19) implies that Next, integrating (4.20) in time, we can obtain from (4.8) that and some positive constant C 14 . Here we have used the Gagliardo-Nirenberg inequality.
Case p = 4α + 2 3 < 1. Then sign(p − 1) = −1 < 0, therefore, by using p < 1 and (3.1), we derive from the Young inequality that  Case p = 1. Using the first equation of (2.1), from integration by parts and applying (1.6), we derive from (3.1) that for some positive constant C 16 , which combined with the Young inequality implies that S Ω n 1−2α ε |∇c ε | 2 + C 16 for all t > 0. On the other hand, due to p = 1 yields to 4α + 2 3 > 2 3 , employing almost exactly the same arguments as in the proof of (4.10)-(4.22) (the minor necessary changes are left as an easy exercise to the reader), we conclude the estimate where γ 0 = min{3α + 1, 2}.

Regularity properties of time derivatives.
To prepare our subsequent compactness properties of (n ε , c ε , m ε , u ε ) by means of the Aubin-Lions lemma (see Simon [27]), we use Lemmas 3.1-4.2 to obtain the following regularity property with respect to the time variable.
With the help of a priori estimates (see Lemmas 4.1-4.3 and 5.1), by extracting suitable subsequences in a standard way (see also [43]), we could see the solution of (1.4) is indeed globally solvable.